Disease ? Bugs?

Those tree notches (where 2 trunks meet) gather all kinds of stuff. Everything just sits there and decays.

Kind of normal for elm (common on Siberian elm) to have some amount of wet wood, or slime flux:
http://extension.psu.edu/pests/plant-diseases/all-fact-sheets/wetwood-or-slime-flux
http://ipm.illinois.edu/diseases/series600/rpd656/
http://shadetreeexpert.com/tree-diseases/
It's a bacterial infection with anaerobic conditions in wounded portions of a tree. The lack of oxygen holds decay in check. If attempts are made to change anaerobic conditions that exist in tree (e.g. drilling drain tubes), then it will cause aerobic conditions that are conducive for fungal wood decay. While unsightly and sometimes stinky, most arborists don't consider it a major problem; however, there are some willing to sell a "fix" for it .

Fell, cut into lengths that will fit into your stove, split, stack in the sun and wind for at least two years, then enjoy the "free" heat that those suckers give off. No more small white worms, smell, or other problems. Feel free to replant future fuel close-by.

I personally don't think that they are any diseases, I have seen trees leaking before. But to be honest they were very small leaks and stopped itself.
I think you should wait some more and get an expert advice before cutting it down.

Lloyd the redneck said:

The only reason I ask is cause the wife is determined we need to cut them down. I don't like to cut them down. Unless nessicary.

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Listen to your wife. I hear here on Hearth it's the only way to a happy life. Plus, who wants all those puss-oozing trees hanging around.